MAT 423 Assignments

Highlighted assignments are due soon.

Due Date

 Assignments
30 Nov Questions 5⋅70, 5⋅73, 5⋅75, 5⋅76, 5⋅83, 5⋅91, 5⋅99 (this problem is in the wrong place; you may use the Isomorphism Theorem to solve it), 5⋅105
2 Dec
  • Questions 5⋅37, 5⋅39, 5⋅41, 5⋅43, 5⋅48, 5⋅57, 5⋅59, 5⋅60, 5⋅63
  • Also compute the left and right cosets of \(\langle\mathbf j\rangle\) in \(Q_8\). Is \(\langle\mathbf j\rangle\) a normal subgroup of \(Q_8\)?
18 Nov
  • Read Section 5⋅4
  • Questions 4⋅43, 4⋅49, 4⋅53, 4⋅55, 4⋅56, 4⋅57, 5⋅25, 5⋅27
    Hints:
    • For 4⋅43, be sure to use the definition of a principal ideal given in class: \(\langle a\rangle=\{ar:r\in R\}\).
    • For 4⋅49(a), I cannot emphasize enough the necessity of looking at some example elements of the ideal. For (i), for instance, what other elements must be in \(\langle3,4\rangle\) besides 3 and 4 themselves? For part (c), Bézout’s Identity can prove very useful — the general Identity, not just the case for \(\gcd(a,b)=1\).
    • For 4⋅55, you may want to use 4⋅53, and for that you probably want to reflect on a point I’ve made over and over in class.
    • For 4⋅56, I’ve already done this, really, though I may have done it for particular polynomials, whereas this is for arbitrary polynomials.
    • That brings us to 4⋅57, probably the hardest of the bunch. It might be a better idea to try and do (a)(ii) before (a)(i), but maybe not. The problem asks you to find the set of elements (called \(\sqrt A\)) such that \(a\) itself might not be in \(A\), but some power of \(a\) is. In terms of (a)(ii), what number is not in \(9\mathbb Z\), but a power of it is? In terms of (a)(i), can you perhaps argue that every element in \(\sqrt A\) is already in \(A\)? Howcome? and so forth.
    • For 5⋅25, use the canonical representation with brackets (\([a]\)) rather than writing out \(a+6\mathbb Z\) each time.
16 NovRead Section 5⋅3
11 Nov
  • Read Section 5⋅2 (we’ve already done 5⋅1, under a different guise)
  • Questions 4⋅8, 4⋅9, 4⋅10, 4⋅11, 4⋅13, 4⋅16, 4⋅21, 4⋅22, 4⋅27, 4⋅30, 4⋅36, 4⋅39
28 Oct
  • Questions 3⋅75, 3⋅76, 3⋅77, 3⋅78, 3⋅83, 3⋅91, 3⋅95, 3⋅96, 3⋅98, 3⋅106, 3⋅113
  • Extra Credit: Questions 3⋅117, 3⋅122, 3⋅123
21 Oct
  • Read the subection on “Evaluating positions in the game” starting on p. 85
  • Questions 3⋅52, 3⋅60, 3⋅68
  • Show that the group \(\Omega_n\) is isomorphic to \(\mathbb Z_n\). Hint: Map generator to generator.
  • Find all the generators of \(\mathbb Z_8\). If it’s isomorphic to \(\Omega_8\) (as shown above), what are the generators of \(\Omega_8\)?
13 Oct
  • Questions 3⋅15, 3⋅17, 3⋅19, 3⋅22, 3⋅23, 3⋅34, 3⋅35, 3⋅43, 3⋅44
    Hint on 3⋅22: Consider \(f=(x-1)(x^2+1)\). If you take the ring of remainders modulo \(f\), you should encounter zero divisors. Identify them, and indicate why the presence of zero divisors shows we are not in a field.
  • Also explain why the polynomial \(x^n+1\) has a root in \(\mathbb Z_2\) for every \(n\in\mathbb Z^+\), so we cannot build a number that is “imaginary and modulo 2” using that polynomial. What about \(x^2+x+1\)?
6 Oct
  • Questions 2⋅53, 2⋅55, 2⋅56, 2⋅58, 2⋅62, 3⋅11
  • Also show that, when we consider \(M=(\mathbb N,+)\) and \(N=(\mathbb N,\times)\) as monoids, we have a function \(f:M\rightarrow N\) defined by \(f(x)=0\) that preserves the operation (i.e., \(f(x+y)=f(x)f(y)\) for all \(x,y\in\mathbb N\)) but not the identity (i.e., \(f(0)\neq1\)).
30 Sep
  • Questions 2⋅26, 2⋅27, 2⋅28, 2⋅38, 2⋅39, 2⋅40, 2⋅41, 2⋅44, 2⋅45, 2⋅46, 2⋅47, 2⋅49
  • Also do these two problems, which somehow didn’t make it into the notes:
    • Is \((B,\vee,\wedge)\) a ring? (Here, \(B=\{T,F\}\), \(\vee\) is Boolean or, and \(\wedge\) is Boolean and. We are asking if \(\vee\) can stand in for addition and \(\wedge\) can stand in for multiplication.)
    • Is \((B,\oplus,\wedge)\) a ring? (Here, \(B=\{T,F\}\), \(\oplus\) is Boolean exclusive or, and \(\wedge\) is Boolean and. We are asking if \(\oplus\) can stand in for addition and \(\wedge\) can stand in for multiplication.)
28 Sep Read Section 3⋅1, preferably 3⋅2 as well
23 Sep Questions 2⋅3, 2⋅4, 2⋅9, 2⋅10, 2⋅11, 2⋅22, 2⋅23, 2⋅24
16 Sep Read Sections 2⋅2 and 2⋅3 of the supplied notes. Questions 1⋅51, 1⋅52, 1⋅53, 1⋅55, 1⋅56, 1⋅57
Extra Credit: 1⋅58
14 Sep Read Sections 2⋅1 and 2⋅2 of the supplied notes.
9 Sep Questions 1⋅34, 1⋅35, 1⋅40, 1⋅44, 1⋅46
If so inclined, Read §1⋅5 and for extra credit do Questions 1⋅47 and 1⋅48
6 Sep Read Section 1⋅6 of the supplied notes.
2 Sep
  • Read Sections 1⋅4 and 1⋅5 of the supplied notes.
  • Questions 1⋅12, 1⋅15, 1⋅17, 1⋅18, 1⋅19
31 Aug Read Sections 1⋅5 and 1⋅6 of the supplied notes.
26 Aug
  • Questions 1⋅1-1⋅6
  • Read Sections 1⋅3 and 1⋅4 of the supplied notes.
24 Aug
  • Read the syllabus. There will be a quiz!
  • Read the Preface and Sections 1⋅1, 1⋅2 of the supplied notes⋅